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TOOL SCALE AND FEATURE SCALE MODELS 13 2 Tool Scale and Feature Scale Models INTRODUCTION Chapter 1, "Industrial Perspectives," outlines the most promising ways to apply plasma modeling and simulation to attack problems that most directly affect plasma tool suppliers and chip manufacturers. The purpose of this chapter is to summarize the current state of plasma modeling and simulation and to identify the most pressing issues that could limit the pace of progress in solving industrially relevant problems. The major industrial problems on which modeling may have an impact involve decreasing the time and cost of developing new plasma tools (shortening the time to market), improving the efficiency of optimizing tool performance to meet changing process objectives, and helping to implement real-time process control into operating tools. Each of these goals implies a somewhat different type of plasma modeling approach. In addition, some plasma models can be developed relatively quickly, while others will take more time. They all share the need to be systematically tested against experimental measurements (validated), and they need to be in a form that can be used conveniently by engineers involved in design, optimization, and control of plasma tools in industry. However, the greatest need for any plasma model is in the area of an improved database for the multitude of elementary processes that collectively determine the physical and chemical dynamics of the system. TOOL SCALE MODELS Over the past decade, modeling and simulation of plasma reactors at the tool scale, meaning the scale of the entire reactor (including the entire wafer), have attracted much attention.1 This is due in large part to the recognition that the problems in the effective application of plasma processes, outlined in Chapter 1, might be fruitfully attacked using modeling and simulation. However, glow discharge plasma simulation is a challenging task, primarily because of (1) the nonequilibrium nature of the plasma, (2) the disparate time scales (< 1 ns to several seconds), and (3) the complexity of the gas phase chemistry and especially the surface chemistry. In an electrical glow discharge, external electromagnetic energy is applied to the system, heating electrons and ions. Since the gases rarely are more than a few percent ionized (at most), electrons experience collisions mostly with much heavier neutral species. The very small mass ratio of the collision partners (electrons/neutral molecules) results in a low efficiency of energy transfer in elastic collisions between electrons and neutral species. Electrons continue to gain energy from the electric fields until their mean energy is sufficiently high to excite inelastic collisions with neutrals. Electron mean energies are typically several electron volts (eV), and the resulting ionization of neutral molecules is the primary sustaining mechanism in a glow discharge. (Note that 1 eV is approximately equivalent in temperature to 11,000 K or 20,000 Â°F). Because of the ambipolar electric fields in the plasma, positive ions are usually accelerated to the boundaries of the plasma and then accelerated further across the sheath potential to impact the surface. Electrons typically diffuse against the confining ambipolar fields to the walls and recombine there with positive ions. In electronegative gases, negative ions form through electron attachment, and are usually lost through gas phase processes such as positive ion-negative ion recombination or electron detachment. Electrons with several eV mean energy can not only ionize molecules, but also can easily dissociate most molecules into fragments. These fragments are the main chemical precursors for both film deposition and etching. Chemisorbed neutral molecular fragments at surfaces, especially when impacted by energetic positive ions from the discharge, are responsible for the chemical reactions that lead to etching. These processes are taking place in chambers with (often) fully
14 DATABASE NEEDS FORMODEUNG AND SIMULATION OF PLASMA PROCESSING three-dimensional spatial variation, and they are in general not at steady state. Modeling this system therefore requires a treatment of 1. Kinetics and transport of charged species (electrons, positive and negative ions); 2. Kinetics and transport of neutral species in the gas phase; and 3. Interaction of charged and neutral species with surfaces. The nonequilibrium nature of the electrons and ions means that these species not only do not have the same average thermal energies, but in addition do not generally share the same form for the distribution of their velocities. Neither electrons nor ions in general follow the Maxwell-Boltzmann form, characteristic of species at local thermal equilibrium. 11ris means that for a rigorous treatment of electron and ion transport some kind of kinetic scheme is necessary: solution of the Boltzmann equation, for example, or a particle simulation such as a Monte Carlo scheme. Also, neutral transport is often complicated by the fact that the collisional mean free path for neutrals (for some low-pressure etching equipment) is on the same order as the dimensions of the reaction chamber. This brings into question the use of continuum or fluid descriptions of neutral transport. An additional complication comes from the fact that neutral species may be in excited vibrational or electronic states. In some cases, then, a rigorous treatment of neutral transport also requires a kinetic approach. In spite of this, schemes that assume some form for distribution functions (i.e. fluid models) have proven useful for conditions under which their limitations are well understood. An increasingly popular approach is to combine POTENTIAL (V) ELECTRON DENSITY (4.0 x 10 11 cm Â·3 ) fluid and kinetic schemes into a hybrid model, in which, ideally, the strengths of both approaches can be combined, while minimizing their weaknesses. In the last 5 years, there have been 1- :c impressive developments in plasma modeling <!) Lii algorithms. These algorithms use fluid, :c kinetic, and hybrid methods to treat plasma and neutral transport and kinetics. Electromagnetic modules have been coupled successfully to the 13.0 6.5 0 6.5 13.0 (a) RADIUS(cm) transport and kinetics codes and have been applied to systems of industrial interest. The availability of engineering workstations with high performance at modest cost has made these developments possible. In addition, E optimizing compilers, convenient "debuggers," ~ "windowing" capabilities, and excellent ~ graphics are widely available and increase the ~ productivity of modelers. An example of a two-dimensional, axisymmetric calculation for an inductively coupled (radio-frequency) 13.0 6.5 0 6.5 13.0 plasma discharge is shown in Figure 2.1. (b) RADIUS (cm) The major missing ingredient in further FIGURE 2.1 An example of a tool-scale simulation of an exploitation of large-scale plasma modeling inductively coupled plasma reactor and various discharge and simulation is the availability of a physical spatial profiles: (top) contour plot of two important discharge and chemical database for a large and diverse characteristics, electron density and plasma potential; set of species present in the discharge, (bottom) plot of electron temperature and the source terms for interacting through a variety of collisional creation of electrons. (Courtesy of Mark Kushner, University processes in the gas phase and at surfaces. In of Illinois at Urbana-Champaign.)